Directed Reading Program Fall 2023

What is it? The Directed Reading Program (DRP) in the UW Madison Department of Mathematics pairs undergraduate students with graduate mentors for semester-long independent studies. During the semester, the student will work through a mathematical text and meet weekly to discuss it with their mentor. The original DRP was started by graduate students at the University of Chicago over a decade ago, and has had immense success. It has since spread to many other math departments who are members of the DRP Network.

Why be a student?

• Learn about exciting math from outside the mainstream curriculum!
• Prepare for future reading and research, including REUs!
• Meet other students interested in math!

Why be a mentor?

• Practice your mentorship skills!
• It strengthens our math community!
• Solidify your knowledge in a subject!

Current Organizers: Ivan Aidun, Allison Byars, John Cobb, John Spoerl, Karan Srivastava

Requirements

At least one hour per week spent in a mentor/mentee setting. Students spend about two hours a week on individual study, outside of mentor/mentee meetings. At the end, students give a 10-12 minute presentation at the end of the semester introducing their topic. This semester, it is scheduled for Wednesday, December 6th.

Applications

Check out our main page for examples of past projects.

Students: Applications are closed.

Mentors: Applications are closed.

Questions?

Contact us at drp-organizers@g-groups.wisc.edu

Projects

Fall 2023 Projects

Title	Abstract	Required Background
Analysis On Graphs	We will study analysis on graphs. More precisely, we will read the book Introduction to Analysis on Graphs by Alexander Grigor'yan. The purpose of the book is to provide an introduction to the subject of the discrete Laplace operator on locally finite graphs. Analysis on graphs is a subject studies classical calculus and differential equations on discrete objects such as graphs. It has connections with many other important areas in math and outside of math.	Students should be familiar with basic calculus (221, 222 and 234) and linear algebra. Analysis courses such as 521 and 522 or basic probability theory are helpful.
Analysis, Geometry, and Combinatorics	I am very flexible and happy to work with your backgrounds and interests. A few things I've thought about are: - Looking at Guth's book "Polynomial Method's in Combinatorics", hopefully culminating in understanding the proof of the finite field Kakeya conjecture. - Measure theory - we'll start with Stein & Shakarchi's book, depending on how that goes, we could move on to Frank Morgan's "Geometric Measure Theory: A Beginner’s Guide" for a bit of geometric measure theory. - Laszlo Lovasz's book "Combinatorial Problems and Exercises". The name is pretty self-explanatory - this books balances more towards problems and less towards theory, but won't require much background. - Spivak's book "Calculus on manifolds". This isn't your usual calculus book - it will introduce differential forms, which are very useful in geometry and analysis.
Fractal geometry	In recent years, a surprising connection between two seemingly unrelated branches of math has emerged. Fractal dimension is a notion of size for sets which, unlike how we normally think about dimension, is not necessarily an integer (e.g. the rough shape of a coastline, the branching of a bolt of lightning, or the chaotic movement of a stock's price have non-integer dimension). Kolmogorov complexity, which measures how much information a string contains, can be applied to characterize some types of fractal dimension. This connection has already yielded impressive new results about fractal sets. In this DRP, we will start by understanding notions of fractal dimension and the definition/properties of Kolmogorov complexity. Then, we'll bridge the gap between the two via "effective" dimension. Effective dimension gives rise to elegant characterizations of both Hausdorff and packing dimension via the recently proven point-to-set principle, which we will study. Finally, we'll see some applications of this alternative definition, which makes it one of the most powerful new tools in fractal geometry.	I would like students who have taken 521-522. But I'm flexible and happy to find something which you feel ready for.
Number theory and Partition Theory	We will investigate modular forms with a view towards algebraic combinatorics and number theory. In this project, we will first read about the interesting connections of modular forms and number theory/arithmetic geometry, with a focus on the theory of theta functions and elliptic curves. We will then transition to a more concrete application of modular forms via partition theory. In particular, we'll cover both analytic and combinatorial techniques, such as the circle method and enumerative combinatorics. We may also dive into more modern applications such as mock theta functions. Depending on participant interest this may turn into a research project based on prior work of the mentor on generalizing the Rogers-Ramanujan identities in partition theory.	Students should have completed a semester of algebra (541) and be familiar with proofs. Familiarity with complex analysis will be useful though not required. Coding knowledge with python would also be useful.
Measure Theory	Euclidean space may seem simple enough, but it contains many complicated subsets with highly counterintuitive properties. Throughout the semester, we'll consider a number of examples of such subsets, including the standard Cantor set, fat and thin Cantor sets, Vitali sets, "large" meager sets, purely unrectifiable sets, Kakeya sets, Nikodym sets, and many more. Readings will be provided and will come from a variety of sources,	Students should either have taken 521 or currently be enrolled in it.
Basics of D-modules	In this project, students will be introduced to D-modules, which allow us to think about ideas from calculus, such as derivatives and differential equations, in an algebraic context. The topics covered will include the Weyl algebra, the Jacobian conjecture, rings of differential operators, modules over the Weyl algebra, and differential equations. The text we will use is "A Primer of Algebraic D-modules" by S.C. Coutinho.	The prerequisite is math 541 or an equivalent: rings, ideals, homomorphisms, polynomial rings, and abstract vector spaces. A basic knowledge of modules is necessary, but this can be introduced at the beginning of the project.
Commutative Algebra	Ideals, Varieties, and Algorithms (Chapters 1-2) with a view toward algebraic geometry.Students are expected to give "mini-talks" on the board for the section we read that week.	Some knowledge of abstract algebra but not much else.
Computability Theory, or, Algebra	1, Computability Theory: we will focus on computability theory, complexity theory, and cryptography. There is a book called Computational Complexity: A Moder Approach, and we can pay more attention to Non-determinism and NP Completeness. 2, Algebra: we will focus on Linear Representations of Finite Groups. It is also the name of a book which describes the correspondence, due to Frobenius, between linear representations and charac­ters. The book says this is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics and have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra.	Strong abilities of Analysis and Discrete Mathematics (including Algebra, etc)
Machine Learning	In this project, students will read/watch parts of the "Practical Deep Learning for Coders" course of fast.ai, a deep learning library that aims to make deep learning as accessible as possible, with the aim of training a basic machine learning model of their own design. In parallel, the students will also develop the skills for some software development practices, such as unit testing, documentation, and key bindings.	Calculus, Linear Algebra, and basic programming experience (preferably including Python) are required. I would also like the students to be mostly independent as designing their own machine learning model is a goal for this project (although this goal is not a hard requirement) and as I might be away for some weeks during the semester.
Dynamics (Analysis/Geometry)	Ergodic theory studies dynamical systems (we can think of this as a repeated action) and their statistical properties (i.e. time averages). For example, consider a circle and pick a point and an angle. Now our action will be repeatedly rotating the point by the chosen angle. Now if we do this action repeatedly for a long time, what will the resulting orbit of the point be? Will it miss part of the circle? Go everywhere? Go everywhere equally often? If we choose an irrational angle the last is true and we call the system ergodic. Ergodic theory is concerned with the study of these systems. While we won’t directly study them, ergodic theory has applications in many other fields -- including number theory, harmonic analysis, probability, and mathematical physics -- making it an incredibly exciting multi-disciplinary area to study. Our plan will be to first spend some time reviewing measure theory, then look at some examples and define ergodicity rigorously. After that we will spend some time studying the various ergodic theorems. If time allows we will extend our study to mixing and entropy.	The Real Analysis sequence (or equivalent) is required. Having seen some measure theory will be extremely helpful though we will spend the first week or two reviewing this topic. Any geometry/topology is useful but not at all required.

Presentation Schedule

Room 1 (Engineering Hall 3349)

Time	Speakers	Title
3:45-4:00	Yikai Zhang & Beining Mu	Computability
4:00-4:45	Aidin Simkin, Yifan Yang, Sena Witzeling	Several Strange Cantor Sets
4:45-5:00	Yancheng Zhu	FastAi in NLP
5:00-5:15	Shi Kaiwen	Machine learning application in stock index prediction
5:15-5:30	Hannah Wang	Trained model for predicting the insurance price

Room 2 (Engineering Hall 3418)

Time	Speakers	Title
3:30-4:00	Aiden Styers, Rui Yu, Yizhou Qian & Anvith Aravati	Laplace Operator on Graphs
4:30-5:00	Benjamin Braiman & Ruoyu Men	Ergodic Theory
5:00-5:15	David Jiang	Ramanujan Partition Identities